Linear representation theory of groups of order 12
From Groupprops
This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 12.
View linear representation theory of group families | View linear representation theory of groups of a particular order |View other specific information about groups of order 12
The list
Degrees of irreducible representations
FACTS TO CHECK AGAINST FOR DEGREES OF IRREDUCIBLE REPRESENTATIONS OVER SPLITTING FIELD:
Divisibility facts: degree of irreducible representation divides group order | degree of irreducible representation divides index of abelian normal subgroup
Size bounds: order of inner automorphism group bounds square of degree of irreducible representation| degree of irreducible representation is bounded by index of abelian subgroup| maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroup
Cumulative facts: sum of squares of degrees of irreducible representations equals order of group | number of irreducible representations equals number of conjugacy classes | number of one-dimensional representations equals order of abelianization
Full listing
| Group | Second part of GAP ID (GAP ID is (12,second part)) | Degrees of irreducible representations as list | Number of irreps of degree 1 | Number of irreps of degree 2 | Number of irreps of degree 3 | Total number of irreducible representations = number of conjugacy classes |
|---|---|---|---|---|---|---|
| dicyclic group:Dic12 | 1 | 1,1,1,1,2,2 | 4 | 2 | 0 | 6 |
| cyclic group:Z12 | 2 | 1,1,1,1,1,1,1,1,1,1,1,1 | 12 | 0 | 0 | 12 |
| alternating group:A4 | 3 | 1,1,1,3 | 3 | 0 | 1 | 4 |
| dihedral group:D12 | 4 | 1,1,1,1,2,2 | 4 | 2 | 0 | 6 |
| direct product of Z6 and Z2 | 5 | 1,1,1,1,1,1,1,1,1,1,1,1 | 12 | 0 | 0 | 12 |
Grouping by degrees of irreducible representations
| Number of irreps of degree 1 | Number of irreps of degree 2 | Number of irreps of degree 3 | Total number of irreps = number of conjugacy classes | Number of groups with these degrees of irreps | Description of groups | List of groups | List of GAP IDs (second part) |
|---|---|---|---|---|---|---|---|
| 12 | 0 | 0 | 12 | 2 | the abelian groups of order 12 | cyclic group:Z12, direct product of Z6 and Z2 | 2,5 |
| 4 | 2 | 0 | 6 | 2 | the dihedral and dicyclic group | dicyclic group:Dic12, dihedral group:D12 | 1,4 |
| 3 | 0 | 1 | 4 | 1 | the alternating group | alternating group:A4 | 3 |
Correspondence between degrees of irreducible representations and conjugacy class sizes
See also element structure of groups of order 12#Conjugacy class sizes.
For groups of order 12, it is true that the list of conjugacy class sizes completely determines the list of degrees of irreducible representations, and vice versa. The details are given below. The middle column, which is the total number of each, separates the description of the list of conjugacy class sizes and the list of degrees of irreducible representations:
| Number of conjugacy classes of size 1 | Number of conjugacy classes of size 2 | Number of conjugacy classes of size 3 | Number of conjugacy classes of size 4 | Total number of conjugacy classes = number of irreducible representations | Number of irreps of degree 1 | Number of irreps of degree 2 | Number of irreps of degree 3 |
|---|---|---|---|---|---|---|---|
| 12 | 0 | 0 | 0 | 12 | 12 | 0 | 0 |
| 2 | 2 | 2 | 0 | 6 | 4 | 2 | 0 |
| 1 | 0 | 1 | 2 | 4 | 3 | 0 | 1 |
Note that the phenomenon of the conjugacy class size statistics and degrees of irreducible representations determining one another is not true for all orders:
